Optimal stability of complement value problems for p-L\'evy operators

Abstract

We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-L\'evy operators, $1 < p < \infty$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-\Delta)^s_p$ with $0 < s \le 1$. If $(-\Delta)^s_p u_s = f_s $ in $\Omega \subset \mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\Omega$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \to 1^-$ in the the optimal, that is, $\|u_s - u_1\|_{W^{s,p}(\Omega)} \to 0$. \smallskip Another subsequent goal underpinning our approach is the robustness of the nonlocal trace spaces; specifically, we also show that the nonlocal trace spaces converge, in an appropriate sense, to the local trace space.